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Mirrors > Home > MPE Home > Th. List > sbcieg | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2129, ax-12 2166. (Revised by GG, 12-Oct-2024.) |
Ref | Expression |
---|---|
sbcieg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcieg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3774 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | sbcieg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | elabg 3662 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
4 | 1, 3 | bitrid 282 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {cab 2702 [wsbc 3773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-sbc 3774 |
This theorem is referenced by: sbcie 3817 2nreu 4443 reuprg0 4708 rabsnif 4729 ralrnmptw 7103 ralrnmpt 7105 fpwwe2lem3 10658 nn1suc 12267 opfi1uzind 14498 mndind 18788 fgcl 23826 cfinfil 23841 csdfil 23842 supfil 23843 fin1aufil 23880 ifeqeqx 32412 nn0min 32668 bnj1452 34811 cdlemk35s 40537 cdlemk39s 40539 cdlemk42 40541 2nn0ind 42505 zindbi 42506 prproropreud 46983 |
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